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Asymptotic estimates of the first eigenvalue of the p-Laplacian
Auteur(s)
Matei, Ana-Maria
Date de parution
2003
In
Advanced Nonlinear Studies
Vol.
2
No
3
De la page
207
A la page
217
Résumé
,We consider a 1-parameter family of hyperbolic surfaces M(t) of genus v which degenerate as t --> 0 and we obtain a precise estimate of lambda(1,p)(t), the first eigenvalue of the p-Laplacian (p > 1) on M(t). In some cases we also give a precise estimate of the first eigenfunctions. As a direct application, we obtain that the quotient (lambda1,q)(1/q)/(lambda1,p)(1/p) (which is invariant under scaling of the metric) is unbounded even on the set of Riemannian manifolds with constant sectional curvature. This is to our knowledge, the first example of a family of manifolds with this property. To prove our results we use in an essential way the geometry of hyperbolic surfaces which is very well known. We show that an eigenfunction for lambda(1,p)(t) of L-p norm one is almost constant in the L-p sense (as t --> 0) on the parts of M(t) with large injectivity radius, and we estimate precisely its p-energy on the parts with small injectivity radius.
Identifiants
Type de publication
journal article
